no, of course Bourbaki was not the first to notice the need. Probably thefirst relevant instance was in Dedekind, who was of the opinion that"nothing is more dangerous in mathematics, than to accept existences withoutsufficient proof" (letter to Lipschitz, 1876; see also his famous letter toKeferstein, 1890). He realized the need for a proposition securing theexistence of an infinite set (proposition no. 66 in his booklet -- not no.6, is this a mistake in Suppes?), the big difference with us is that hethought he could prove it by purely logical means. Dedekind seems to havebeen motivated by Bolzano's "Paradoxien des Unendlichen" (1851) in givinghis proof, and probably also in realizing the need for it.

Hilbert, Russell, and many others seem to have accepted Dedekind's proof.But after the publication of the Russell paradox in 1903, widely publicizedin the books of Russell and Frege, criticism of Dedekind began to appear.One of the first instances is in Hilbert's paper 'On the foundations oflogic and arithmetic', published in 1905 (included in van Heijenoort, "FromFrege to Gödel", 1967). Subsequently, the proposition was transformed intoan axiom by Hilbert's colleague Ernst Zermelo. In Zermelo's axiomatizationpaper of 1908 (also in van Heijenoort), he calls it "Dedekind's axiom" as away of acknowledging the importance of that precedent.

The axiom systems for set theory became popular during the 1920s and 1930s.Also within other systems usual at the time (type theory), people adoptedexplicitly an axiom of infinity -- see e.g. the famous papers onmathematical logic by Gödel (1931) and Tarski (1933 and 1935). There werealso interesting philosophical discussions about this topic, e.g. by Ramsey.By the time of Bourbaki's presentation, it was only too well known that oneneeded this axiom.

Best wishes to all, and peace on earth,

Jose Ferreiros

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Several weeks ago Alexander Zenkin (alexzen@com2com.ru), in a post to HMwhich unfortunately I did not keep, said that Bourbaki stated the need for Axiom of Infinty: There exists an infinite set

Was this argument first made by Bourbaki?

In Patrick Suppes _Axiomatic Set Theory_ 2nd (?) edition New York: DoverThe attempt to prove the existence of an infinite set of objects has arather bizarre and sometimes tortured history. Proposition No. 6 ofDedekind's famous Was sind und was sollen die Zahlen?, first published in1888, asserts that there is an infinite system. (Dedekind's systemscorrespond to our sets.)[footnote] A similar argument is to be found in Bolzano [Paradoxien desUnendlichen, Liepsiz 1851] section 13<end quote>